Fuzzy Opinion Networks: A Mathematical Framework for the Propagation of Opinions and Their Uncertainties Across Social Networks

Abstract

Human opinions are inherently fuzzy, because the carrier of opinions is natural language where fuzzy words and vague expressions are the mainstays rather than exceptions, thus a good mathematical theory for opinion dynamics should consider the opinion and its uncertainty simultaneously.We propose a new mathematical framework for the evolution and propagation of opinions and their uncertainties, called Fuzzy Opinion Network (FON), which is the connection of a number of Gaussian Nodes, possibly through some weighted average, time-delay or logic operators, where a Gaussian Node is a Gaussian fuzzy set with the center and the standard deviation being the node inputs and the fuzzy set itself being the node output. In this framework an opinion is modeled as a Gaussian fuzzy set with the center representing the opinion itself and the standard deviation characterizing the uncertainty about the opinion. We study the basic connections of Fuzzy Opinion Networks, including basic center, basic standard deviation (sdv), basic center-sdv, chain-in-center and chain-in-sdv connections, and we analyze a number of dynamic connections, including self-feedback, compromising husband with persistent wife (or vice versa), compromising with each other, ring connection with state-dependent uncertainty, “smart” student versus “stubborn” student, and bounded confidence connection, to show how opinions and their uncertainties propagate and evolve across different network structures and scenarios.The main insights gained from the mathematical analyses of the FONs include:(a) Opinion leaders are important for a community, in the sense that the anxiety (uncertainty) of all members in the community may easily go to infinity if everyone compromises fully with all others (i.e., if there is no opinion leader in the community); when there is an opinion leader (a stubborn node in the FON), however, the anxieties of all members in the community will converge to a finite number;(b) Speed matters for time-varying FONs; for example, depending on the speed of gaining confidence through communicating with others, the uncertainties of the members in the community may stabilize at a finite number if the speed is fast or go to infinity if the speed is slow; and,(c) For state-dependent connections such as the Ring Connection or the Bounded Confidence Connection studied in this paper, a consensus will be reached if a central agent collects the opinions of all individuals and discloses them back to each individual (this shows that the media is a powerful machine to control people’s opinion), while in the decentralized control scenario (people know only the opinions of neighbors), different communities emerge where people in the same community reach a consensus but these different consensuses remain separated forever (this explains why people in the same country tend to share similar opinions while people at different countries usually have different opinions).